Measuring Capital Stock in Germany - Slightly abridged version of a paper published in the journal Wirtschaft und Statistik 11/2006
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NATIONAL ACCOUNTS
Dr. Oda Schmalwasser, Michael Schidlowski
Measuring Capital Stock
in Germany
Slightly abridged version of a paper published
in the journal Wirtschaft und Statistik 11/2006
This treatise describes calculations of fixed assets or capi- mations up to the calculation of assets and liabilities, which
tal stock and consumption of fixed capital in the context of conclude the system of accounts with balance sheets. Fixed
national accounts in Germany. To start with, we explain the assets or capital stocks are in the focus of the calculation of
terms used, followed by an overview of fixed assets and con- assets and liabilities and the way they change is the most
sumption of fixed capital estimations, and then we present important indicator of the extent to which production activ-
the Perpetual Inventory Method as the principal method ity increases national wealth and creates, at the same time,
used. We explain both the mathematical model applied in the conditions needed to continue the production of goods
Germany and the computational implementations based on and services in future. Unlike the United States, which
Excel and Visual Basic for Applications. The treatise ends determines fixed assets in net terms only as required for
with a description of the data base and the results of capi- balance sheet calculations, Germany has pursued an inte-
tal stock and consumption of fixed capital estimations that grated approach of determining fixed assets in both gross
were obtained in the context of the 2005 revision of national and net terms since the early 1970s, when capital stock cal-
accounts and the 2006 backward calculations. culations started. That means that fixed assets are deter-
mined not only in net terms as the current value of the capi-
Preliminary note tal stock, which is important for balance sheet calculations,
but also in gross terms as a measure needed for the analy-
Gross domestic product is the central value of national sis of productivity and incorporating the capital stock’s pro-
accounts. In Germany the Federal Statistical Office deter- duction capacity.
mines it by way of reconciling production and expenditure
approach data. As data on business profits are not available, As the main focus of capital stock examinations has been
it is not possible to use the income approach in order to on fixed assets in net terms, productivity analysis at inter-
determine gross domestic product or gross national income. national level has basically relied on the “capital services”
In contrast, the United States uses the income approach as concept in recent years.2) The term “capital costs” (“Kapi-
a central method to calculate gross domestic product and talkosten”), introduced in German, correctly includes the
gross national income. Work is in progress there to elabo- financing and distribution aspect, but does not cover the
rate a self-contained production approach.1) The setting of production or value added aspect, which has been read into
different priorities regarding the determination of central it concerning its recommended use for productivity exam-
values in national accounts is a common feature of all esti- inations. This treatise shows that conventional German
1) See Lawson, A./Moyer, B./Okubo, S./Planting, M.: “Integrating Industry and National Economic Accounts” (http://www.oecd.org/dataoecd/6/41/33871180.ppt; as of: 7 November 2006)
and the paper by Moyer, B./Planting, M./Fahim-Nader, M./Lum, S., on which this presentation was based at the OECD Working Party meeting on national accounts in 2004: “Preview of
the Comprehensive Revision of the Annual Industry Accounts: Integrating the Annual Input-Output Accounts and the Gross-Domestic-Product-by-Industry Accounts” (http://www.oecd.org/
dataoecd/42/18/33785748.pdf; as of: 7 November 2006).
2) See OECD (ed.): “Measuring Capital”, OECD Manual, 2001.
Statistisches Bundesamt • Wirtschaft und Statistik 1NATIONAL ACCOUNTS
fixed assets calculations represent an integrated approach each accounting period. The accounts that show stocks are
to the calculation of both the capital stock’s current value, called balance sheets. Stocks are also recorded for popula-
on the one hand, and of the capacities available for pro- tion and employment. However, these stocks are recorded
duction purposes, on the other hand, in a way that is con- as mean values over the accounting period.”4) Stock of fixed
sistent with the formats in which the gross domestic prod- assets means a holding of assets, it therefore is a value mea-
uct is displayed. That is why this treatise places emphasis sured at a point in time. “Flows refer to actions and effects
mainly on the coherent presentation and explanation of the of events that take place within a given period of time, while
Federal Statistical Office’s methods of calculation used to stocks refer to positions at a point of time.”5)
determine fixed assets or capital stocks. In conclusion, we
shall explain the database used and the results obtained by A combination of data on stocks and flows is methodolog-
the calculations made in the context of the 2005 revision ically problematic, because stocks of fixed assets relate to
of national accounts for Germany regarding the years 1991 a moment of time, whereas flow data refer to a period of
to 2004 and of the 2006 backward calculation for the years time. If we wish to calculate capital productivity similar to
1970 to 1991 relating to the former federal territory. labour productivity, we must make sure that the other pro-
duction factor is subject to the same procedure as is laid
down for employment (and the population) in the ESA-95
1 Definition of fixed assets (see above): We must generate an annual average. Average
annual gross fixed assets at constant prices are traditionally
and capital stock referred to as Kapitalstock in Germany’s national accounts.
Kapitalstock calculations provide us with the methodologi-
The terms “Kapitalstock” and “fixed assets” are often used cally correct tools needed to determine capital productivity
synonymously in German and they are also used synony- as gross domestic product per unit of capital stock, capital
mously with the English term “capital stock”. A definition output ratios as their reciprocal value and capital intensity
of the terms used is imperative, as we are faced with con- as capital stock per employed person (as an annual aver-
cepts, such as (stocks of) fixed assets in gross and net age). Although international concepts do not provide for
terms apart from different price concepts, “gross” and “net such calculations, they are, nevertheless, methodologically
capital stock” as well as “wealth” and “productive capital exact. However, the German word “Kapitalstock”, which is
stock”. It is the only way to avoid ambiguities caused by dif- used to denote this annual average, often gives rise to mis-
ferent meanings assigned to one and the same concept. understandings caused by confusion with the English term
Microeconomic issues are not considered here. The defini- “capital stock”, which means a stock figure related to a
tions are based on the internationally agreed concepts of point of time.
national accounts. They provide the definitions and explan-
atory methodological notes that statistical offices use as a However, neither the SNA nor the ESA use a term such as
basis for their measurement concepts. Accordingly, (stocks “capital stock”. Depreciation is exactly referred to as “con-
of) fixed assets comprise all produced fixed assets that sumption of fixed capital” and investments as “gross fixed
are used repeatedly, or continuously, in processes of pro- capital formation”, but stocks do not provide for a term such
duction for more than one year. They include tangible and as “capital stock”, but only for “stocks of fixed assets”. The
intangible assets. Tangible fixed assets consist of the stock reason is that according to the SNA and the ESA not all gross
of machinery and equipment, residential and non-residen- fixed capital formation actually increases fixed assets.6)
tial buildings, and cultivated assets. Intangible fixed assets Both major improvements to non-produced non-financial
comprise mineral exploration, computer software and large assets and the costs of ownership transfer on non-pro-
databases as well as entertainment, literary or artistic orig- duced non-financial assets as part of gross fixed capital for-
inals. mation are referred to as additions to the value of non-pro-
duced non-financial assets. The name says it already: This
In order to characterise capital stocks and fixed assets it part of gross fixed capital formation does not increase the
is very important to distinguish between flows and stocks, value of fixed assets, but adds to the value of non-produced
a distinction made both by the 1993 System of National non-financial assets (in particular, land). This approach may
Accounts (SNA) of the United Nations and by the 1995 Euro- be reasonable from the perspective of fixed assets, but it
pean System of Accounts (ESA). “Flows reflect the creation, has the undesirable side effect that gross fixed capital for-
transformation, exchange, transfer or extinction of eco- mation, fixed assets and consumption of fixed capital are
nomic value. They involve changes in the value of … assets defined differently: Fixed assets do not include additions to
or liabilities.”3) The gross domestic product and most of the the value of non-produced non-financial assets, i. e. addi-
other national accounts variables are flows that relate to a tions, which are included in gross fixed capital formation
period of time. “Stocks are holdings of assets and liabilities and need to be written off as consumption of fixed capital.
at a point in time. Stocks are recorded at start and end of Now the “capital stock” concept used by the OECD manual
3) Annex A to Council Regulation (EC) No. 2223/96 of 25 June 1996 on the European System of National and Regional Accounts in the European Community (Official Journal of the EC No.
L 310, p. 1), section 1.32.
4) Ibid., section 1.47.
5) Ibid., section 1.31.
6) A detailed comparison of the classification of fixed assets and gross fixed capital formation can be found in Schmalwasser, O.: „Revision der Anlagevermögensrechnung 1991 bis 2001“ in
WiSta 5/2001, p. 344.
2 Statistisches Bundesamt • Wirtschaft und StatistikNATIONAL ACCOUNTS
on “Measuring Capital”7) establishes a connection between of fixed capital: consumption of fixed capital in the context
gross fixed capital formation and consumption of fixed cap- of calculating domestic product, including charges equiva-
ital by applying exactly the same definition to the different lent to the consumption of accumulated costs of ownership
types of assets, which are subject to capital consumption transfer on unimproved land, on the one hand, and in the
(see chart 1). Coverage and classification of capital stock context of the calculation of assets and liabilities exclud-
by type of asset, considered in this manual, show that they ing the latter ones, on the other hand. With the 2005 revi-
are composed of fixed assets, on the one hand, and major sion of national accounts and anticipating a revision of the
improvements to non-produced non-financial assets and SNA, this duplicate approach was abandoned and accumu-
costs associated with the transfer of ownership on non-pro- lated costs of ownership transfer on unimproved land were
duced non-financial assets as part of non-produced (tangi- included in fixed assets as well. Strictly speaking, Germa-
ble) assets, on the other hand.8) ny’s fixed assets are now equivalent to capital stock in terms
of the OECD manual. We have deliberately accepted a slight
“Capital stock”, in German “Kapitalstock”, is used in this deviation from the ESA concepts presently in force, because
treatise in the same way as in the OECD definition of “capi- the cost of ownership transfer on land is a relatively insignif-
tal stock”. The traditional German “Kapitalstock” described icant component of total fixed assets.
above is additionally specified as “annual average”, which
has also been done for some time in the Federal Statistical
Office’s English translations of publication tables in order to 2 Overview of the calculation of
avoid misunderstandings.
fixed assets and consumption
Judging on the present course of discussions, the revi- of fixed capital
sion of the SNA 1993, which is to be completed by 2008,
is expected to put an end to the practice that different defi-
nitions are applied to gross fixed capital formation and con- 2.1 Gross and net approach and
sumption of fixed capital, on the one hand, and fixed assets, consumption of fixed capital
on the other hand. Then all parts of gross fixed capital for-
mation will be included in fixed assets as well and all assets The way in which capital stock estimations are presently
actually produced will be covered as produced assets. With organised in Germany arises from the traditional approach
Germany’s transition to the ESA-95, the costs of ownership of the Federal Statistical Office9) to such calculations and
transfer on unimproved land were not considered to be con- from the requirements of the data transmission programme
stituent parts of fixed assets, whereas land improvements, under the ESA 1995.10) Besides the obligatory transmission
as in the past, were recorded under buildings. Two different of fixed assets data, a point of great importance is the sup-
approaches to the calculation of fixed assets and consump- ply of data on consumption of fixed capital. Consumption
tion of fixed capital were used to calculate total consumption of fixed capital in national accounts measures the amount
Chart 1: Definition of gross fixed capital formation, fixed assets, consumption of fixed capital and capital stock
based on the SNA and ESA classification
Gross fixed capital formation Fixed assets Consumption of fixed capital Capital stock
P.51 AN.11 K.1 (OECD Manual)
P.511
AN.111 To be calculated with respect to
Acquisitions less disposals of tangible Tangible fixed assets
Tangible fixed assets tangible fixed assets
fixed assets
P.512
AN.112 To be calculated with respect to
Acquisitions less disposals of Intangible fixed assets
Intangible fixed assets intangible fixed assets
intangible fixed assets
P.513
Additions to the value of non-
produced non-financial assets, of
which ________ To be calculated with respect to major Major improvements to and costs of
P.5131 improvements to and costs of ownership transfer on non-financial
Major improvements to non-financial ownership transfer on non-financial non-produced assets as parts of non-
non-produced assets non-produced assets produced assets
P.5132
Costs of ownership transfer on non-
financial non-produced assets
Component of AN.2 non-produced assets
7) See footnote 2.
8) Ibid., p. 23 and p. 25 f.
9) See Lützel, H.: „Das reproduzierbare Anlagevermögen in Preisen von 1962“ in WiSta 10/1971, p. 593 ff., and Lützel, H.: „Das reproduzierbare Sachvermögen zu Anschaffungs- und zu Wie-
derbeschaffungspreisen“ in WiSta 11/1972, p. 611 ff.
10) See Annex B of the ESA regulation, footnote 3.
Statistisches Bundesamt • Wirtschaft und Statistik 3NATIONAL ACCOUNTS
of fixed assets used up during the period under consider- The above distinction is required, since the capital stock is
ation as a result of normal wear and tear and foreseeable composed of individual pieces of fixed assets, which were
obsolescence including a provision for losses as a result of acquired in different years and which can be summed up in
accidental damage which can be insured against. This cal- different ways. Using a perpetual inventory method which is
culation is done for all fixed assets, including both tangi- based on the (original) acquisition prices that were actually
ble fixed assets and intangible fixed assets such as min- paid in a given year of investment, one speaks of historic
eral exploration costs and software. Animals are excluded acquisition prices or historic cost value, which best agree
by definition. Land improvement costs are not separately with an enterprise’s bookkeeping data. This method has the
determined in Germany. They are included in buildings and disadvantage that fully identical fixed assets are likely to be
structures. Costs of ownership transfer on unimproved land included in the accounts with completely different acquisi-
are written down as consumption of fixed capital together tion prices, just because these goods were acquired in dif-
with the respective types of buildings. That is why calcula- ferent years and purchaser’s prices changed between these
tions of fixed assets and consumption of fixed capital form years. The result of this calculation at historic cost for differ-
a unity. ent years of acquisition has nothing to do with the current
prices commonly used for flows, which relate to one report-
According to the ESA transmission programme, the stock of ing year, even if the calculation as such is based on cur-
fixed assets is to be recorded both in gross terms and net rent prices of gross fixed capital formation. In order to con-
terms, in either case at current replacement costs and con- vert the stock at historic acquisition prices into one year’s
stant replacement costs. Registration at historic cost value prices, we would need to make sure that the yearly propor-
is voluntary and done in line with the valuation principles of tions of fixed capital formation related to the total stock
microeconomic and fiscal balance sheets. Gross approach of assets are available in a detailed breakdown by type of
means that fixed assets are recorded as part of fixed assets assets and that the respective price indices are obtainable
at full value (current market prices of new fixed assets of a as well. This would allow us to construct a weighted average
corresponding quality) without consideration of the decline based on the price indices for all fixed assets acquired in all
in an asset’s value by the time when these fixed assets years, which would be much too expensive a procedure to
retire from the production process. This is done in consid- be used. The Federal Statistical Office has abandoned this
eration of the fact that the production process makes use price concept for capacity reasons. Although this concept
of all fixed assets irrespective of their age and that the out- complies with the valuation techniques applied by busi-
put achieved is more or less the same every year, provided nesses’ bookkeeping systems, one should note that the val-
that fixed assets are maintained and repaired regularly. For ues recorded for fixed assets at historic acquisition prices in
instance, a 500 MW power generating unit produces these net terms are not comparable, because the methods used to
500 MW as long as it is connected to the power grid even calculate consumption of fixed capital are very different.
after 20 years, when most of its value will have been written
off as consumption of fixed capital. That is why the stock of
gross fixed assets is a suitable value for the analysis of pro- Before the Perpetual Inventory Method is used in capital
duction processes. stock calculations, gross fixed capital formation data relat-
ing to different years are brought into a comparable form,
In balance sheets, however, fixed assets are to be valued which is done by using series of investment data at constant
at market prices (current market value) at the time to which prices. As a result we obtain fixed assets data at constant
the balance sheet relates. Since, in most cases, there are no prices of one reporting year, i. e. at constant replacement
market prices for existing fixed assets of different age other prices (costs). The data on the stock of fixed assets defined
than motor vehicles, the net approach provides for a deduc- in that way and on the related decline in value by way of con-
tion of the amount that has been written off as consump- sumption of fixed capital comply with the concept of con-
tion of fixed capital since the date of investment.11) This is stant prices applied in national accounts, which means that
a good compromise that reconciles what is desirable in the- the entire stock of all fixed assets is valued at prices of one
ory with what is feasible in practice. Accordingly, the net reference year or at base-year prices. To obtain the current
capital stock corresponds to the current value of the fixed replacement prices (costs) of a given year under consider-
assets measured in terms of the fixed assets’ current mar- ation, we revaluate the stock, which was consistently val-
ket value. ued at base-year prices, into prices of the respective year
under consideration with respect to all years. All that now
is still needed is data on the price indices of the year under
2.2 Price concepts consideration (related to the base year), broken down into
detailed items of fixed assets. Hence, current replacement
Three price concepts are used in fixed assets calculations: prices (costs) for fixed assets correspond to current prices
for flows in national accounts, since all fixed assets are val-
– current replacement costs,
ued at prices of that one year under consideration as are all
– constant replacement costs, and other flow data, so that we have a real match. This relates, in
particular, to the data on consumption of fixed capital at cur-
– historic cost value. rent prices, which are determined by using this method. The-
11) See ESA 1995, sections 7.29 and 7.33.
4 Statistisches Bundesamt • Wirtschaft und StatistikNATIONAL ACCOUNTS
oretically, it would even be possible to revaluate all series of uct and gross national income. The output of other non-mar-
gross fixed capital formation data for every year under con- ket producers is determined as total production costs and
sideration into the prices of one specific year under consid- consumption of fixed capital is one component of these
eration and to use the Perpetual Inventory Method later on, costs.12) That is why we need to determine consumption of
but that is an extra effort we spare by calculating the prices fixed capital in the required detail by industry and govern-
for one base year only and revaluating the results for all ment subsector, i. e. federal (central) government (S.1311),
years under consideration. state government (S.1312), local government (S.1313) and
social security funds (S.1314), in order to arrive at gross
The transition from constant prices to previous year’s value added of other non-market producers. In addition, we
prices with chaining in national accounts brought about a need consumption of fixed capital of market producers for
methodological problem in the calculation of fixed assets, the sector accounts of the government sector by subsector,
since price changes are correctly recorded only in relation to of non-profit institutions serving households, financial cor-
the previous year, but not in relation to the reference year. porations (S.12) by subsector, non-financial corporations
Constant prices, in their turn, correctly reflect the changes in (S.11) and households (S.14) as well as total consumption
prices between the reporting year and the base-year of price of fixed capital for the sectors in total.
registration, but they are of restricted use for the measure-
ment of year-to-year changes in prices, because weighting That means that calculations of fixed assets and consump-
varies. Since the constant price approach corresponds to tion of fixed capital must have a multidimensional structure.
the approach used to cumulate gross fixed capital forma- Major dimensions such as industries, type of fixed assets
tion, constant prices continue to be used for the calculation and sectors/subsectors are shown in fig. 1. As a first step
of stocks of fixed assets and consumption of fixed capital. this presentation starts with a cross-classification by indus-
On this basis it is even possible to generate data at previous try and type of fixed assets. In Germany we need not make
year’s prices by applying the price indices that were used to an extra distinction between market and non-market pro-
revaluate data into current or replacement prices. For that ducers, because this differentiation is completely available
purpose we must additionally convert the values at constant – with one exception – in terms of industries and subsec-
(replacement) prices into values at previous year’s prices, tors (see section 4.1, chart 2). It should be noted that by far
which is done by using the price indices of the previous not all fields of the multidimensional matrix are filled with
year. This method is actually used to calculate consumption data. But for every field that is filled with data we need to
of fixed capital as a flow item. However, if the stock of fixed have long series of data on gross fixed capital formation and
assets is related to a concrete date, we are faced with addi- a good’s service life (distribution), which varies over time
tional methodological problems caused by the prices that (see chapters 3 and 4). That means that two more dimen-
must also relate to that date. For that reason we do not pres- sions are added to the calculations.
ently show the stock of fixed assets by chaining year-to-year
data. 2.4 Perpetual Inventory Method and
That is why, in order to show capital productivity, we use elements of the direct method
data on gross domestic product or gross value added at cur-
rent prices and average capital stock for a year at current As no comprehensive direct data is available on the stock
replacement prices. Relating chained data on gross domes- of fixed assets in Germany, the main method of calculating
tic product or gross value added to a year’s average capital fixed assets and consumption of fixed capital is the Perpet-
stock at constant replacement prices is not a methodologi- ual Inventory Method (PIM), which is recommended by the
cally perfect solution. However, taking into account that the ESA. The details of this method are explained in chapter 3.
structure of fixed assets does not change considerably from In addition, we also use elements of the direct method in
year to year, the difference between constant prices and Germany:
chained prices is practically negligible. That means that we
– The calculation of cultivated assets is based on annual
can even tolerate such an approach, although it is not per-
data of agricultural statistics on livestock and cultivated
fect from a methodological point of view.
areas. This data on stocks and areas is valued, using the
average weights and prices of a given calendar year (cur-
2.3 Multidimensionality of calculations rent replacement costs) and those of the year 2000 (con-
stant replacement costs), which are provided by agricul-
The ESA transmission programme provides for a cross clas- tural accounting. According to the ESA-1995 livestock is
sification of fixed assets by 31 industries and three types of not written off, so that in this case net assets are equal to
fixed assets. Much more information is required – directly gross assets.
or indirectly – with respect to consumption of fixed capital.
This is mainly due to the fact that consumption of fixed cap- – Information on the stocks inherited from the ex-GDR is
ital by other non-market producers of general government based on annual fixed assets reports and reports on the
(sector S.13) and non-profit institutions serving households material, technological and age structure of the GDR’s
(S.15) has an influence on the size of gross domestic prod- fixed assets, i. e. on direct statistical records about the
12) See also ESA 1995, section 3.53.
Statistisches Bundesamt • Wirtschaft und Statistik 5NATIONAL ACCOUNTS
Fig. 1
Multidimensionality of fixed assets and consumption of fixed capital calculations
Sector S. 11
Assets AN 1.1…
Sector S. 12
1 2 ... an ... AN
1 AN 1,1,1 ... AN 1,an,1 ... AN 1,AN,1
2
. . . .
. . Fixed assets. .
Sector s
. . . .
AN 1.1
Industries (A60)
Sector S
a AN a,1,1 ... AN a,an,1 ... AN a,AN,1
. . . .
. . . .
. . . . AN a,an,s
60 AN 60,1,1 ... AN 60,an,1 ... AN 60,AN,1
AN a,an,S
stock of fixed assets. These stocks were separately de- are recorded as other changes in the volume of fixed as-
termined on this basis, but comprehensive adaptations sets. Various cases of special retirements (withdrawals
were required, since what we needed for the calculation from the stock before the end of assumed service life)
was only the part that was still in use for production pur- are recorded for fixed assets under this item, including
poses after transition from planned to market economy, the repercussions they have on consumption of fixed
rather than the entire stock of the ex-GDR’s fixed assets. capital and retirements resulting from PIM, described in
For example, assets that proved to be useless under mar- section 4.3.
ket conditions by the end of 1992 were excluded from
the initial stock of inherited fixed assets, determined as Accordingly, the stock of fixed assets in gross and net terms
of 1 July 1990. The stocks of fixed assets inherited by the recorded at start of 1991 for Germany as a whole consists of
new federal states as of the beginning of 1991 were ad- the following items:
ded to the initial stocks obtained by using the Perpetual
Inventory Method. Write-offs and retirements were calcu- Stock of fixed assets at start of 1991, obtained by using
lated well in advance and must annually be taken into the perpetual inventory method
account for inherited stocks.13)
+ Stock of cultivated assets at start of 1991
– The ESA-1995 stipulates that changes caused by excep-
tional incidents, which do not comply with the original + Stocks of inherited fixed assets of the ex-GDR at start of
assumptions made by the perpetual inventory method, 1991
13) On the inclusion of inherited stocks from the former GDR see section 1.2.3 in Schmalwasser, O., footnote 6, here: p. 347.
6 Statistisches Bundesamt • Wirtschaft und StatistikNATIONAL ACCOUNTS
– Accumulated special retirements and their counterpart We must also exclude the respective East-German data
entries for the former federal territory until and including from the stock of cultivated assets at start of 1991, and we
1990 must take into account that the data on the stock at start of
year, which were obtained by using the perpetual inventory
= Stock of fixed assets at start of 1991 for Germany as a method, already include the new federal states’ fixed capi-
whole tal formation for the second half of 1990 (after the introduc-
tion of DM on 1 July 1990). That means that we must exclude
Consumption of fixed capital and retirements are deter- the respective parts of fixed capital formation, when calcu-
mined analogously, which is shown here using consump- lating the stocks for the former federal territory at start of
tion of fixed capital as an example: 1991. Accordingly, we must take into account the resulting
consumption of fixed capital for 1990 in order to determine
Consumption of fixed capital obtained by using the per-
the 1990 consumption of fixed capital for the former fed-
petual inventory method
eral territory. The influence on retirements in 1990 can be
+ Consumption of old fixed capital which is further used by neglected. On that basis we effect a “real” backward calcu-
the new federal states lation of the former federal territory’s fixed assets as shown
below, using the gross stock as an example:
– Counterpart entries to consumption of fixed capital cor-
responding to other changes in the volume of fixed as- Gross stock of fixed assets for the former federal territory
sets at start of year
= Total consumption of fixed capital – Additions to the previous year’s fixed assets
Then the flows connected with fixed assets, which were + Retirements from the previous year’s fixed assets
determined in this way, and the stocks of fixed assets for
Germany relating to the start of 1991 are used as the basis = Gross stock for the former federal territory at start of pre-
on which the stocks of fixed assets are determined for the vious year
years to come as follows:
Gross stock at start of year 3 Use of the Perpetual Inventory
+ Additions to fixed assets Method in Germany
– Retirements of fixed assets 3.1 Brief outline of the
= Gross stock at end of year (= gross stock at start of the Perpetual Inventory Method
following year)
As outlined in section 2.4, the Perpetual Inventory Method
In the calculation of gross assets, entries for special retire- (PIM) is the main method used to determine fixed assets
ments in conjunction with other changes in the volume of and consumption of fixed capital in Germany. This approach
assets are recorded in the other changes in the volume of is based on the idea that today’s stock of fixed assets is
assets account according to the basic pattern described composed of assets that were added to the stock at some
above and are included in retirements. Net assets, however, time in the past. Taking into account the service life of fixed
require other changes in the volume of assets to be consid- assets, we can calculate the percentage of assets added in
ered separately, because these changes cannot be recorded past years, which are still operational in the stock at start of
as special consumption of fixed capital: the year under consideration, and determine the time when
these assets will be withdrawn (retire from the stock). If the
Net stock of assets at start of year depreciation method is applied, we can use this informa-
tion to derive data on consumption of fixed capital for every
+ Additions to fixed assets reference period. The use of the Perpetual Inventory Method
requires the following conditions to be fulfilled: (1) data
– Consumption of fixed capital series on gross fixed capital formation are available, which
reach far back into the past, (see section 4.1) and (2) it is
– Other changes in the volume of assets (net)
possible to estimate the average service life for the various
= Net stock of assets at end of year (= net stock at start of types of fixed assets (see section 4.2).
the following year)
The average (economic) service life is estimated for all
Backward calculations of fixed assets and consumption goods which are subject to consumption of fixed capital. This
of fixed capital for 1970 – 1991 regarding the former fed- is done, assuming normal wear and tear and foreseeable
eral territory are also based on the stock at start of 1991. obsolescence as a result of technological progress. The esti-
For that purpose we determine the stock at start of 1991 mation also covers the risk of losses due to insurable acci-
for the former federal territory according to the basic pat- dental damage. As it is unrealistic to assume that all goods
tern described above. In so doing, we must leave out of with the same average service life and added in the same
account not only the inherited, old stocks of the former GDR. year will also retire at the same time, we use a mortality
Statistisches Bundesamt • Wirtschaft und Statistik 7NATIONAL ACCOUNTS
Fig. 2
Calculation of fixed assets and consumption of fixed capital
using the Perpetual Inventory Method
Results Basic material Results
Time series on gross fixed capital formation
at constant prices (long-run data of the past
and broken down by group of assets)
(+) Density Average Depreciation (+)
Gross fixed function economic method (in Net fixed
assets (value (e.g. gamma service life most cases Consumption assets (current
(+) as new) at
Retirements (+)
at constant distribution, (estimated by straight line, of fixed capital market value)
constant prices1) lognormal group of geometrically at constant at constant
replacement distribution) assets) digressive) prices1) replacement
costs1) costs1)
(–) (–)
Distribution Distribution of consump-
of retirements tion of fixed capital
Annual average prices indices
Gross fixed Retirements (price indices of gross fixed capital formation) Consumption Net fixed
assets (value at current of fixed assets (current
as new) at prices capital at market value)
current (current Price indices at the beginning of year (estimated) current prices at current
replacement replacement (replacement replacement
costs costs) Other changes in the volume of costs) costs
assets at constant prices
Value as new (gross) Current market value (net)
1) The respective data on historic cost value can be derived from the time series on gross fixed capital formation at current prices.
Statistisches Bundesamt 2006 - 01 - 1344
(retirement) function to distribute retirements so that they volume of assets (see section 4.3). This item enables us
scatter around the average service life in a roughly bell- to ensure that exceptional, unanticipated events, which
shaped manner (see section 3.2). have an impact on the value of fixed assets and, as a
later consequence, on the size of consumption of fixed
Fig. 2 schematically outlines the use of the perpetual inven- capital, are included in the calculation.
tory method for the calculation of fixed assets and consump-
tion of fixed capital. This description clarifies the essential
3.2 Mathematical model
points of the method used:
The mathematical model used to implement the perpetual
– To begin with, we need some basic information, includ- inventory method in Germany is designed so that the flow
ing long-term series of data on gross fixed capital forma- data, i.e. retirements and consumption of fixed capital, can
tion (at constant prices for calculations at constant and directly be derived from the two pieces of basic information
current prices), approaches to service life determination needed for the calculation of fixed assets, i.e. gross capi-
and corresponding price indices for the various fixed as- tal formation and service life of the various fixed assets. To
sets needed for calculations at current price, provided describe the model14), we use the following standardised
that the depreciation method and the distribution of reti- symbols:
rements are taken for granted.
i : investment year
– Consumption of fixed capital is calculated in a self-con-
tained way, which does not require fixed assets to be cal- t : (current) reporting year
culated in an intermediate step (“inner circle”).
n : service life in years
– Any deviations from the normal economic trend that this
model imputes can be recorded as other changes in the ñ : average service life
14) See footnote 9.
8 Statistisches Bundesamt • Wirtschaft und StatistikNATIONAL ACCOUNTS
Ii : additions of year i The retirements of the reporting year t are obtained as the
sum of assets added during the past years i and having a
Ii,n : additions of year i with service life n service life of n = t – i years:
fñ(n) : mortality function of gross fixed capital formation
At = ¦ I i • f n~ n ( t −i )
i 0, p > 0 in the stock and which have already been calculated by way
of the mortality function, we can directly calculate consump-
tion of fixed capital without having to determine the size of
Parameters a and p determine, inter alia, the steepness of
stocks. Due to the proportions of retirements already con-
the mortality function. The curve is the steeper, the higher
sidered we obtain a non-linear function for the depreci-
these dilation parameters are. Dilation parameter 9 was
ation curve, as fig. 4 shows, displaying the distribution of
assumed to be suitable for most goods. It ensures results
consumption of fixed capital with respect to goods having
most closely approximating past evaluations regarding
an average service life of 10 or 20 years. Although straight
retirements of vehicles. Adopting parameter 9 as the normal
line depreciation is used, the data obtained as a result of
case, we obtain a mortality function as follows:
distributing consumption of fixed capital approximate those
−
9n obtained by way of geometric depreciation, which is used,
~ −9 n8e
f n~ ( n ) = 9 9 ( 8! )−1 n ñ inter alia, by the United States. This results from the fact
that straight line depreciation follows the real service life of
Fig. 3 displays the curve progression of a mortality function the assets, for which consumption of fixed capital is mea-
related to an average service life of 10 or 20 years. sured, whereas according to the mortality function some of
Fig. 3 Fig. 4
Mortality function Distribution of consumption
for assets with an average service life of of fixed capital for assets with an average service
10 or 20 years life of 10 or 20 years
Retirements as a percentage Retirements as a percentage Consumption of fixed capital as a Consumption of fixed capital as a
of an investment year of an investment year percentage of an investment year percentage of an investment year
14 14 12 12
12 12
10 10
10 Years
10 10
8 8
10 Years
8 8
6 6
6 6 20 Years
20 Years 4 4
4 4
2 2
2 2
0 0 0 0
0 10 20 30 0 10 20 30 40
Service life in years Service life in years
Statistisches Bundesamt 2006 - 01 - 1345 Statistisches Bundesamt 2006 - 01 - 1346
Statistisches Bundesamt • Wirtschaft und Statistik 9NATIONAL ACCOUNTS
the goods that were acquired in an investment year were Fig. 5
already withdrawn from the stock after a short span of time
(long before reaching the average service life of the respec- Basic structure of the file system
tive group of assets).
Distribution of service lives, retirement distribution
and distribution of consumption of fixed capital
We obtain a reporting year’s consumption of fixed capital
for assets added in a given investment year by multiplying
the respective share of depreciation with the value of gross
fixed capital formation. The following applies to consump-
tion of fixed capital in the reporting year t related to the
Calculation of the retirement
additions of the year i: distribution and distribution
of consumption of fixed capital B-matrix 1 B-matrix 2
Di , t = I i ¦ dt ( n ) • f n~ ( n )
n≥t −i
Consumption of fixed capital in the reporting year t is ob- was always used in subsequent calculations of the gamma
tained as the sum of consumption of fixed capital for the distribution. Fig. 6 shows the structure of one such service
various years of fixed capital formation: life approach.
Dt = ¦ Di , t Fig. 6
i≤t
Example of a service life approach to machinery and equipment
3.3 IT solution
The software system for the perpetual inventory method
was originally programmed on a mainframe platform in the
late 1970s. At the end of the 1990s, when the shutdown of
the system was imminent, we had to think about alterna-
tive IT tools that could be used to handle the complex com-
putations required. A problem of the mainframe variant was
the scarcity of available storage space, which had the effect
that it was necessary to redetermine the distributions of
retirements and consumption of fixed capital in every com-
putational cycle and delete them again afterwards. Storage
space ceased to be a problem, when modernised and larger
hard disk systems were introduced at PC level. Considering
that the spreadsheet programme Excel was generally avail-
able for difficult mathematical computations, a solution was
chosen, which was based on “Visual Basic for Applications
(VBA)”.
Another problem that had to be considered was the extent
to which it would be possible to map the model in the file
system taking into account the model’s complexity. As the
computation of the gamma distribution and its final pro-
gression (linearization of the asymptotic function, see
below) needs very much computing power, this process was The table header includes the investment years and the pro-
separated from the rest of the file system. Besides, a strict portions of the respective assets grouped by service life are
separation was made in the file system between service life listed thereunder for each investment year. As additional
approaches and the calculation of flows and stocks, inter- information the average service life for the year in question
nally referred to (briefly) as B-matrices. This appeared advis- was indicated at the end of the column. These service life
able, first of all, in order to prevent large files from placing files still consist of several Excel worksheets, which include
an extremely high burden on the network. This trichotomy – both the mortality distribution and the ensuing distribution
service life approaches, computation of the gamma distri- regarding consumption of fixed capital. But the computing
bution, and B-matrices – is schematically displayed in fig. 5 process as such has been relocated to another file. If a ser-
and will be explained in more detail below. vice life distribution changes, computing is done using VBA
so that the new distribution is transferred to the computing
In compiling the service life approaches and their goods file and, on completion of the computing process, the distri-
proportions a presentation was chosen, which it would butions of retirements and consumption of fixed capital are
be easy to understand. This did not raise any problems as returned to the service life distribution. A separate mortal-
regards machinery and equipment, as dilation parameter 9 ity distribution is calculated for each individual service life
10 Statistisches Bundesamt • Wirtschaft und StatistikNATIONAL ACCOUNTS
group. Subsequently, the various mortality distributions so For each section, which is subject to computation, a sepa-
defined are weighted with the proportions of the compos- rate file is used to determine flows and stocks. Fig. 8 shows
ite asset groups related to fixed capital formation of the year the structure of the B-matrices files.
in question.
A factor that has to be considered as regards the computing Fig. 8
process as such is the fact that the gamma distribution is an
infinite function. That is the reason why we discontinue the Structure of a B-matrix
gamma distribution in Germany as soon as 99 % of assets
have been withdrawn. In the years that follow the data is
updated using the previous year’s proportion as long as all
assets are actually withdrawn from the stock. The related
computing process is much more challenging in terms of
mathematics and programming than the gamma distribu-
tion as such. VBA and if-queries were used to implement
this step of computation.
Whilst the computing model of machinery and equipment
was largely suitable for intangible assets as well, it was nec-
essary to choose a different approach to buildings, because
here it was not always possible to use just one standard dila-
tion parameter in the gamma distribution. For example, the
parameters chosen for residential buildings were different;
the dilation parameter 7 was laid down for new buildings
and 13 for modernisation activities. Fig. 7 shows the typi-
cal structure of a table displaying the service life approach
to buildings.
Fig. 7
Example of a service life approach to buildings and structures All flows (consumption of fixed capital, retirements) and
stocks (in net and gross terms at start of year) of fixed assets
estimations are calculated in these B-matrices. Exceptional
factors of influence (special retirements) are not considered
by these files. Linking is used to apply the distribution of the
mortality and depreciation functions to the service life dis-
tribution of the section, subject to computation. This model
provides very many possibilities of ad hoc analysis. On the
one hand, gross capital formation can be subject to changes
and their influence on other flows becomes visible at once.
On the other hand, one can directly see how the calculation
will change by simply modifying the service life distribution,
which results in corresponding changes regarding the distri-
bution of the mortality and depreciation functions.
4 Data bases for calculations in the
Unlike the described model for machinery and equip- context of the 2005 revision and
ment, this approach provides for all assets to be arranged
in triads. The first cell of this group represents the dilation the 2006 backward calculation
parameter, the second one the average service life and the
third one the proportion of that asset for the year in ques- 4.1 Long-term series of gross
tion. Here again, a VBA procedure is started, which transfers fixed capital formation
service life data to the buildings worksheet used for com-
puting purposes, and afterwards the ensuing distributions Gross fixed capital formation means additions to fixed
of retirements and consumption of fixed capital are trans- assets or the capital stock. If it is not possible or not desir-
ferred back to the service life file. German fixed assets cal- able to use benchmark estimates for the capital stock in
culations include 150 service life distributions for the vari- a start year, the series of gross fixed capital formation,
ous computing areas, but only two files for computing the on which the perpetual inventory method is based, must
gamma distribution and the cut-off process of this function. include a very long period of the past.15) Since, according
15) See also OECD Manual, footnote 2, p. 44.
Statistisches Bundesamt • Wirtschaft und Statistik 11NATIONAL ACCOUNTS
to the gamma function, retirements may oscillate by more 4.2 Service life approaches
than double around the average service life, data on gross
fixed capital formation should be available at least for a cor- A category which is to be determined for all series of data
respondingly long period of the past. The internal series on gross fixed capital formation is average (economic) ser-
of gross fixed capital formation in Germany start for build- vice life – the second major input needed for the perpetual
ings in 1799, for machinery and equipment in 1899 and for inventory method. Service life is the period, during which
intangible assets in 1945 or even later. an asset’s consumption is recorded. This value is insecure,
because it is geared to the future. In order to determine it in
Gross fixed capital formation is broken down by line with the ESA-1995, it is necessary to take into account
normal wear and tear as well as economic obsolescence
– more than 200 types of machinery and equipment, and losses of fixed assets as a result of accidental damage
8 kinds of buildings and four types of other fixed assets, which can be insured against. It is assumed that the assets
are correctly maintained and minor repairs are continuously
– 60 industries and made. Service life approaches are determined by breaking
down gross fixed capital formation by type of asset in as
– institutional sectors of national accounts (five sectors much detail as possible. The 2005 revision brought about
and seven sub-sectors) minor modifications in the assumptions on the assets’ aver-
age service lives, as they were adapted to the new asset
(please, also refer to fig. 1). That means that gross fixed structures.
capital formation is available in three classification profiles.
Regarding the consumption of fixed capital calculation, it The most important basis for the determination of an asset’s
is also necessary to distinguish between market producers average service life regarding machinery and equipment
and other non-market producers in gross fixed capital for- and some buildings is provided by the AfA tables, which
mation of the government sector (S.13) and non-profit insti- are issued by the Federal Ministry of Finance. These tables
tutions serving households (S.15) (see chart 2), because contain deeply specified assets together with the service
consumption of fixed capital of other non-market pro- life data to be used in order to determine consumption of
ducers of the government sector and of non-profit insti- fixed capital for tax purposes. As the determination of an
tutions serving households has a direct influence on the asset’s fiscal life is based on the assumption that individual
size of gross domestic product and gross national income businesses act according to the principle of precaution, an
(see section 2.3). Although gross fixed capital formation is asset’s real life is, as a rule, longer than its fiscal life. That
already available in a highly detailed manner, it may some- is why the figures of the AfA tables are augmented by an
times be necessary to specify it with even more detail for average 20 to 100 %, when determining the average (eco-
the purpose of fixed assets and consumption of fixed capi- nomic) service life for the purpose of national accounts. The
tal calculations (see fig. 6 and fig. 7). information used for that purpose was provided by enter-
prises and associations (expert assessments). Service life
data of equal asset groups is sometimes differentiated by
Chart 2: Industries in which general government (S.13) and/or industry. For example, lorries operated in construction are
non-profit institutions serving households (S.15) are involved supposed to have shorter service lives than those operated
Sectors in other industries. An internal cross-classification of inves-
A601) Industries S.11/S.12/
tors (assets/industry) is used to determine service life by
S.13 S.15 industry.
S.142)
01 Agriculture, hunting . . . . . . . . . . MP MP
02 Forestry . . . . . . . . . . . . . . . . . . . . . . MP MP The service lives of buildings and structures, in particu-
41 Collection, purification and lar residential buildings, commercial buildings and public
distribution of water . . . . . . . . MP MP buildings, were extracted from the long-term accounts with-
63 Supporting and auxiliary
transport activities . . . . . . . . . MP MP/NMP out major modifications. Many resources were opened by
70 Real estate activities . . . . . . . . . MP MP MP the latest revision when these assets had to be written off
73 Research and development . . . MP NMP NMP and/or recorded as fixed assets for the first time in order to
75 Public administration, defence;
compulsory social security . NMP determine the service life of roads and other structures of
80 Education . . . . . . . . . . . . . . . . . . . . MP NMP NMP general government and fixed assets used by the mili-
85 Health, veterinary and social
work . . . . . . . . . . . . . . . . . . . . . . . MP NMP NMP
tary. These resources included, for instance, investigations
90 Sewage and refuse disposal . . MP MP made by the German Institute for Economic Research (DIW
91 Activities of membership Berlin) and the ifo Institute for Economic Research e. V. as
organisations n.e.c. . . . . . . . . MP NMP
92 Recreational, cultural and
well as texts on how to determine the value of real estate
sporting activities . . . . . . . . . . MP NMP NMP and the cost directive elaborated by the Federal Ministry of
MP: Market output; NMP: Other non-market output
Defence.16)
1) The A60 of national accounts corresponds to the two-digit items of the
industrial classification, issue 2003 (WZ 2003). – 2) Non-financial and The latest revision also tapped various resources to deter-
financial corporations and households. mine the service life of intangible assets. Mineral explo-
16) See Schmalwasser, O., footnote 6, here: p. 349.
12 Statistisches Bundesamt • Wirtschaft und StatistikNATIONAL ACCOUNTS
ration data could be obtained from AfA tables. The service longed in 2001. But not every change in the fiscal service
life of software is differentiated by two groups. Software for life necessarily has an impact on the economic service life
mainframes is supposed to have a longer service life than (according to an estimation of national accounts). Fiscal
PC software. In the course of the 1980s the proportion of service lives are just one source among many others, which
PC software grew continuously. The proportions of the two serve as an indication of possible changes occurring in eco-
types of software were also differentiated by industry. The nomic service lives. According to the estimation of national
average life of literary and artistic originals was estimated accounts, the economic service life is also changed, for
on the basis of differentiated information on motion pic- instance, on the basis of expert assessments and plausi-
tures, TV productions, sound storage media, music compo- bility checks. But structural effects may occur even if the
sitions, artistic performances and texts.17) service life approaches in the most detailed breakdown by
asset do not change at all. That means that every investment
Average lives of assets are obtained by type of asset, indus- year has a service life distribution of its own. Chart 3 shows
try and sector for every investment year. The service life the average service lives by type of asset and the respec-
estimates by type of asset are revised every ten to 15 years, tive span of time on which a (sub) group’s average service
i. e. in relatively large intervals. This was based on several life is based.
considerations. In the last decades of the past century Afa
service lives were repeatedly shortened and partly pro- A distinction must be made between the average service
lives for the various investment years, on which the calcu-
lations are based, and the average service life of assets
Chart 3: Average service lives and range of average service lives
within the group by type of assets for the 2000 investment year in the stock. The latter one corresponds to the reciprocal
in years value of the depreciation rate (rate of consumption of fixed
capital) for the total economy. It is obtained as a result of
Range of average the division of gross annual average stock of fixed assets by
Average service
Type of asset
life
service lives by consumption of fixed capital, if possible, in a breakdown by
group
type of asset. The average for the total economy depends on
Buildings and structures . . . . . . . . . . . . . 66 15 – 150 the composition of fixed assets by goods that have very dif-
Dwellings . . . . . . . . . . . . . . . . . . . . . . . . . . 74 40 – 95 ferent service lives (in particular buildings and structures,
Roads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 35 – 116 machinery and equipment, and intangible assets).
Other structures of general
government . . . . . . . . . . . . . . . . . . . . . 47 25 – 150
Non-residential buildings of general
government . . . . . . . . . . . . . . . . . . . . . 66 25 – 68
4.3 Consideration of special influences
Other non-residential buildings . . . . 53 15 – 100
According to the ESA-1995, changes in the volume of assets
Machinery and equipment (according
to CPA)1) . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 – 30 and liabilities caused by extraordinary events deviating from
Transport equipment . . . . . . . . . . . . . . 11 8 – 25 the perpetual inventory method’s original assumptions are
Motor vehicles, trailers and semi- recorded as other changes in the volume of assets. This
trailers (34) . . . . . . . . . . . . . . . . . . . . 9 8 – 15
Other transport equipment (35) . . 21 12 – 25
item includes those special retirements of fixed assets that
were covered by German fixed assets calculations already
Other machinery and equipment . . . 12 5 – 30
Fabricated metal products (28) . . 18 14 – 22 in the past. It relates, in particular, to the reduction of refin-
Machinery and equipment n.e.c. ery capacities, iron and steel capacities and ship-building
(29) . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 8 – 30 (shipyard crisis) and to corrections of the gross stock, when
Office machinery and computers
(30) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 – 9
used ships are sold abroad.18) Furthermore, from 1993 on
Electrical machinery and apparatus this item has been used to record special retirements of old
n.e.c. (31) . . . . . . . . . . . . . . . . . . . . . 18 8 – 22 assets from the ex-GDR which are no longer usable from an
Radio, television and economic point of view, provided that these assets were
communication equipment and
apparatus (32) . . . . . . . . . . . . . . . . . 10 5 – 17 not already excluded when determining the initial stocks
Medical, precision and optical as of 1 July 1990. Correction is also required in the case of
instruments, watches and clocks repeated sales of real estate within short intervals of time: It
(33) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 10 – 22
Furniture, other manufactured is assumed that real estate is sold at market prices and that
goods n.e.c. (36) . . . . . . . . . . . . . . 16 8 – 30 it is not possible, as a rule, to realise the full costs associ-
Other machinery and equipment ated with the transfers of ownership (tax on purchase of real
(part of 17 – 27) . . . . . . . . . . . . . . . 13 7 – 20
estate, fees of brokers, notaries and courts). The respective
Cultivated assets
amount should be considered for buildings under special
Vineyards . . . . . . . . . . . . . . . . . . . . . . . . . 20
Hop fields . . . . . . . . . . . . . . . . . . . . . . . . . 15 retirements caused by costs associated with the transfers
Asparagus fields . . . . . . . . . . . . . . . . . . . 8 of ownership. And, finally, the high losses of fixed assets,
Fruit trees plantations . . . . . . . . . . . . . 10 which were caused by the flood in August 2002, were
Intangible assets . . . . . . . . . . . . . . . . . . . . 5 5 – 30 also recorded in this category as catastrophic losses. The
1) Statistical goods classification in conjunction with the industries (kinds sources used for that purpose consisted of lists of damages
of activity) used in the European Economic Community, 2002. compiled by the Federal Government and by Sachsen (Sax-
17) See also Frankford, L.: „Urheberrechte in den Volkswirtschaftlichen Gesamtrechnungen“ in WiSta 5/2000, p. 320 ff.
18) See Schmidt, L.: „Reproduzierbares Anlagevermögen in erweiterter Bereichsgliederung“ in WiSta 5/1986, p. 499 ff., here: p. 503 f.
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